2平面问题基本理论

1、 2 Theory of Plane Problems21 Plane Stress and Plane Strain22 Differential Equations of Equilibrium23 Geometrical Equations. Rigid-body Displacement24 Physical Equations25 Stress at a Point26 Boundary Conditions. Saint-Venants Principle27 Solution of Plane Problem in Terms of Displacements28 Solutio

2、n of Plane Problem in Terms of Stresses29 Case of Constant Body Forces. Stress Function*SOLUTION OF PLANE PROBLEM*For the solution of an elasticity problem, we can proceed in three different ways:1. Take the displacement components as the basic unknown functions, formulate a system of differential e

3、quations and boundary conditions containing the displacement components only, solve for these unknown functions and thereby find the strain components by the geometrical equations and then the stress components by the physical equations. 2. Take the stress components as the basic unknown functions,

4、formulate a system of differential equations and boundary conditions containing the stress components only, solve for these unknown functions and thereby find the strain components by the physical equations and then the displacement components by the geometric equation.3. Take some of the displaceme

5、nt components and also some of the stress components as the basic unknown functions, formulate a system of differential equations and boundary conditions containing the stress components only, solve for these unknown functions and thereby find the other unknown functions.27 按位移求解平面问题平面问题的基本未知量有x，y，x

6、y，x，y，xy，u，v，根据基本方程即可求解。Solution of Plane Problem in Terms of Displacements求解方法有：按位移求解；按应力求解；混合求解 P26按位移求解：以位移分量为基本函数，由只含位移分量的微分方程和边界条件求出位移分量后，再求其他的未知量。Take the displacement components as the basic unknown function，formulate a system of differential equations and boundary conditions containing the d

7、isplacement components only，solve for these unknown functions and thereby find the strain components by the geometrical equations and then the stress components by the physical equations. 导出按位移求解的微分方程和边界条件 1、微分方程 (differential equations) Formulate the differential equations and boundary conditions f

8、or the solution of a plane problem in terms of displacements 几何方程：geometrical equations物理方程（平面应力问题）physical equations（plane stress problem）将几何方程代入物理方程Substitution of geometric Eqs. into these equations将上述方程代入平衡微分方程Now, using these relations in equilibrium equations按位移求解的平衡微分方程 拉密方程在平面问题中的应用The diffe

9、rential equations for the solution of the problem in terms of displacements. 2、边界条件(Boundary conditions) lx+mxy=Xmy+lxy=Y平面问题的应力边界条件（Stress boundary conditions of a plane problem）按位移求解时的应力边界条件为：用位移表示的应力边界条件。We obtain the stress boundary conditions of the problem in terms of displacements.归纳(To sum u

10、p), 按位移求解平面问题，要使位移分量满足拉密方程和边界条件，求出位移后，可用物理方程求应力，用几何方程求变形。The displace components u(x,y), v(x,y) in a plane stress problem must satisfythroughout the body considered and also satisfyon the surface of the body.For a plane strain problem, it is necessary in above equations. 对平面应变问题，只许将上述方程中的 ；将 即可。28 按

11、应力求解平面问题 相容方程 P28Solution of Plane Problem in Terms of Stresses按应力求解：以应力分量为基本函数，由只含应力分量的平衡微分方程和相容方程及边界条件求出应力分量后，再求其他的未知量。Take the stress components as the basic unknown function，formulate a system of differential equations and boundary conditions containing the stress components only，solve for these

12、 unknown functions and thereby find the strain components by the physical equations and then the displacement components by the geometrical equations.含应力分量，需保留The two differential equations of equilibrium contain the stress components only and may be used for their solution. 需建立补充方程 相容方程The third di

13、fferential equation can be obtained from the geometrical and physical equations . 相容方程（变形协调方程） Compatibility equation 1、平面问题的几何方程The geometrical equations of a plane problem are2、将x对y的二阶导数和y对x的二阶导数相加 Adding the second derivative of x with respect to y and the second derivative of y with respect to x

14、, we get 相容方程Compatibility equation应变分量x、y和xy必须满足这个方程，才能保证位移分量u，v的存在。 若所选的x、y和xy不满足这个方程，那么，由几何方程中的任意两个所求出的位移分量，将不满足第三个方程。例如选x=0，y=0，xy=cxy 不满足相容方程由此应变求位移The compatibility equation for strain must be satisfied by the strain components x, y and xy to ensure the existence of single-valued continuous fu

15、nctions u and v connected with the strain components by the geometrical equations.第三个方程不能满足，所求u，v不存在 用应力表示的相容方程 Compatibility equation in terms of strainBy using physical equations, the compatibility equation can be transformed into a relation between the stress components. 将物理方程代入 平面应力问题 for a plan

16、e stress problemSubstitution of the physical equations into简化上述相容方程（利用平衡方程） To transform this equation into a different form more suitable for use, we eliminate the term involving xy by using the differential equations of equilibrium. 用应力表示的相容方程the compatibility equation in terms of stresses.Differe

17、ntiating the first equation with respect to x and the second with respect to y, adding them up and noting that xy=yx, we get(将两式分别对x及y求导，并相加得)(将其代入相容方程，并简化后，得)Substituting this into the compatibility equation and performing some simplification, we obtain平面应变问题的相容方程Compatibility equation in terms of

18、strainCompatibility equation for a plane strain problemFor a plane strain problem, an equation similar to above equation may be obtained simply byThe result is 归纳：1、按应力求解平面问题，要求应力分量必须满足平衡微分方程和相应的相容方程，在边界上还要满足应力边界条件（P29）。In the solution of a plane problem in terms of stresses，the stress components mu

19、st satisfy the differential equations of equilibrium and compatibility equation in the case of place strain. Besides,they must satisfy the stress boundary conditions.2、由于位移边界条件无法用应力分量或其导数来表示，所以对位移边界条件或混合边界条件，不可能按应力求解得出精确解。Since the displacement boundary conditions can be expressed neither in terms o

20、f stress components nor in terms of their derivatives with respect to the coordinates,displacement boundary problems and mixed boundary problems cannot be solved in terms of stresses.对应力边界问题，应力分量满足了平衡微分方程、相应的相容方程和应力边界条件，其应力分量就能完全确定？在多连体中，要完全确定位移分量，还必须利用“位移须为单值”这个条件No，还必须考虑弹性体是否单连体In the solution of

21、elasticity problems，it is necessary to distinguish between simply connected bodies and multiply connected ones.多连体：有两个或两个以上连续边界的物体，如：有孔口的物体单连体：只有一个连续边界的物体simply connected bodies: an arbitrary closed curve lying in the body can be shrunk to a point,by continuous contraction,without passing outside it

22、s boundaries. Otherwise,the body will be said to be multiply connected.In the case of multiply connected body, there might be some arbitrary functions leading to multi-valued displacements, which are impossible in a continuous body. Then, we have to consider the condition of single-valued displaceme

23、nts to determine the stresses.In plane problems, however, we may also briefly define a simply connected body as one with only one continuous boundary and a multiply connected body as one with two or more boundaries.29 常体力情况下的简化 体力不随坐标而变化（重力、惯性力） 应力分量应满足： （a）（b）Case of Constant Body ForcesIn many eng

24、ineering problems,the body forces are constant.On the condition of constant body forces, the compatibility equations will reduce to the homogeneous differential equation 上述方程中不含材料常数，所以对两类平面问题都适用.Now the differential equations of equilibrium and the stress boundary conditions, as well as the compatib

25、ility equation, do not contain any elastic constant and are the same for both kinds of plane problems.只要弹性体（单连体）边界相同，外载相同，不管是何种材料，也不管是平面应力状态或平面应变状态，应力分布是相同的（位移及变形是否相同？）In a stress boundary problem for a simply connected body with a certain boundary and subjected to certain external forces, the stres

26、s components will have the same distribution in both plane condition.This conclusion is very helpful in the experimental analysis.（1）可将某种材料，某种状态下所求的应力分量的结论用于其他材料或其他状态（边界条件，外荷载相同）We may use any model material convenient for stress measurement instead of the structure material on which the measurement

27、 might be impossible.（a）（2）在实验中，可以用便于测量的材料来制造模型；或用平面应力情况下的薄板来代替平面应变情况下的长柱体.We may use a model in plane stress condition (a thin slice) instead of one in plane strain condition (a long cylindrical body).（b）The stress components are determined by the differential equations:（a） is nonhomogeneous and, t

28、herefore, its general solution may be expressed as the sum of a particular solution and the general solution of the homogeneous system考察（a），其解由非齐次方程的特解+齐次方程的通解 特解设为：x=-Xx，y=-Yy，xy=0 （c）或 x=0，y=0，xy=-Xy-Yx或 x=-Xx-Yy，y=-Xx-Yy，xy=0只要能满足方程即可取（c）式求齐次方程的通解将方程变为(1)(2)满足（1），必存在一个函数A（x，y）使得：According to diff

29、erential calculus, for (1), there exists a certain function A(x，y) so that：Rewrite同理满足（2），必存在一个函数B（x，y）使得：So Similarly, (2) ensures the existence of another function B(x，y) so that：必存在一个函数（x,y），且Which ensures the existence of still another function (x, y) so that所以，齐次方程的通解为：We obtain the general sol

30、ution of homogeneous equations：平衡微分方程的解为：（c）Now, the superposition of the general solution with the particular solution yields the following complete solution:The function （x,y） is known as the stress function for plane problems, or the Airys stress function.（x,y）称为平面问题的应力函数艾瑞应力函数With any function （

31、x,y）,the stress components so defined always satisfy the differential equations. This function （x,y）is known as the stress function or the Airys stress function. 应力分量除满足平衡微分方程外，还必须同时满足相容方程，所以将（c）代入相容方程In order for the stress components to satisfy the compatibility equation as well,the stress functio

32、n must satisfy a certain equation.Or This is the compatibility equation in terms of the stress function . 2(Xx)= 2(Yy)=0 in the condition of constant body force 此方程为双调和方程，写为：or be simply written as：若不计体积力，即 X=0；Y=0When body forces are not considered, the solution will reduce toThus：in the solution o

33、f plane problems in terms of stresses, when the body forces are constant, it is only to solve for the stress function from the single differential equation and then find the stress components byBut these stress components must satisfy the stress boundary condition. In the case of multiply connected

34、bodies, the condition of single-valued displacements must be inspected in addition.归纳：按应力求解平面问题时，如果体力是常量，则由 求解应力函数，然后按求应力分量，这些应力分量在边界上满足应力边界条件，在多连体中，还须考虑位移单值条件.To solve the partial differential equations of elasticity together with the given boundary conditions, the direct method of solution is usua

35、lly impossible. We have to use the inverse method or the semi-inverse method.In the inverse method, some functions satisfying the differential equations are taken and examined to see what boundary conditions these functions will satisfy and thereby to know what problems they can solve. In the case of solution by Airys stress function, we select some function satisfying the compatibility equation, find the stress components, and then find the surface force components. In this way, we identify the problem which the stress function can solve.In the semi-i